Set theory and its applications. L. Babinkostova, A. E. Caicedo, S. Geschke, M. Scheepers, eds. Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011. ISBN-10: 0-8218-4812-7 ISBN-13: 978-0-8218-4812-8

Here is a link to the AMS page for it, a link to its table of contents, and the preface:

The Boise Extravaganza in Set Theory (BEST) started in 1992 as a small, locally funded conference dedicated to Set Theory and its Applications. A number of years after its inception BEST started being funded by the National Science Foundation. Without this funding it would not have been possible to maintain the conference. The conference remained relatively small with many opportunities for its participants to meet informally. We like to think that during these years BEST has made it possible for the numerous set theorists who have participated in it to absorb, besides the new developments featured in the conference talks, also part of the folklore and traditions of the ﬁeld of set theory and its relatives. An explicit effort was made to bring together role models from various career stages in set theory as well as the new generation to support some notion of continuity in the ﬁeld.
This volume has a similar purpose. In it the reader will ﬁnd valuable papers ranging from surveys that put in print here set theoretic knowledge that has been around for several decades as unpublished lore, to hybrid survey-research papers, to pure research papers. Readers can be assured of the authority of each paper since each has been carefully refereed. The reader will also ﬁnd that the subjects treated in these papers range over several of the historically strongly represented areas of set theory and its relatives. Rather than expounding the virtues of each paper individually here, we invite the reader to learn from the authors.
Bringing to publication such a collection of papers is not possible without the generous dedication of authors and referees and the services of a publisher. We would like to thank all authors and referees for their selﬂess contributions to this volume. And we particularly would like to thank the publisher, Contemporary Mathematics, and Christine Thivierge, for the guidance they provided during this process.

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Nice sharing.
Many k-12 and undergraduate students think that Set theory is a basic thing made for children and has no application in advanced mathematics.
But in fact the set theory is the basis of the whole of mathematics.

Georgii: Let me start with some brief remarks. In a series of three papers: a. Wacław Sierpiński, "Contribution à la théorie des séries divergentes", Comp. Rend. Soc. Sci. Varsovie 3 (1910) 89–93 (in Polish). b. Wacław Sierpiński, "Remarque sur la théorème de Riemann relatif aux séries semi-convergentes", Prac. Mat. Fiz. XXI (1910) 17–20 […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter does not imply the existence of a Vitali set. More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E_0$ is a successor cardinal to ${ […]

Marginalia to a theorem of Silver (see also this link) by Keith I. Devlin and R. B. Jensen, 1975. A humble title and yet, undoubtedly, one of the most important papers of all time in set theory.

Given a positive integer $a$, the Ramsey number $R(a)$ is the least $n$ such that whenever the edges of the complete graph $K_n$ are colored using only two colors, we necessarily have a copy of $K_a$ with all its edges of the same color. For example, $R(3)= 6$, which is usually stated by saying that in a party of 6 people, necessarily there are 3 that know e […]

Equality is part of the background (first-order) logic, so it is included, but there is no need to mention it. The situation is the same in many other theories. If you want to work in a language without equality, on the other hand, then this is mentioned explicitly. It is true that from extensionality (and logical axioms), one can prove that two sets are equ […]

$L$ has such a nice canonical structure that one can use it to define a global well-ordering. That is, there is a formula $\phi(u,v)$ that (provably in $\mathsf{ZF}$) well-orders all of $L$, so that its restriction to any specific set $A$ in $L$ is a set well-ordering of $A$. The well-ordering $\varphi$ you are asking about can be obtained as the restriction […]

Gödel sentences are by construction $\Pi^0_1$ statements, that is, they have the form "for all $n$ ...", where ... is a recursive statement (think "a statement that a computer can decide"). For instance, the typical Gödel sentence for a system $T$ coming from the second incompleteness theorem says that "for all $n$ that code a proof […]

When I first saw the question, I remembered there was a proof on MO using Ramsey theory, but couldn't remember how the argument went, so I came up with the following, that I first posted as a comment: A cute proof using Schur's theorem: Fix $a$ in your semigroup $S$, and color $n$ and $m$ with the same color whenever $a^n=a^m$. By Schur's theo […]

It depends on what you are doing. I assume by lower level you really mean high level, or general, or 2-digit class. In that case, 54 is general topology, 26 is real functions, 03 is mathematical logic and foundations. "Point-set topology" most likely refers to the stuff in 54, or to the theory of Baire functions, as in 26A21, or to descriptive set […]

Nice sharing.

Many k-12 and undergraduate students think that Set theory is a basic thing made for children and has no application in advanced mathematics.

But in fact the set theory is the basis of the whole of mathematics.

Are the BEST conferences still going on?

Hi Aaron. I expect so. We skipped a year.